Duality Methods for Topological Phases Guo Chuan Thiang University of Adelaide Institute for Geometry and its Applications Recent Progress in Mathematics of Topological Insulators 06/09/18 1 / 27
Outline ◮ I will discuss how Poincar´ e duality and T-duality can be used to understand topological phases in new ways. ◮ Bulk-boundary correspondence is an “index-theoretic” idea — that boundary zero modes (analysis) detect bulk topology. ◮ So it is natural to Poincar´ e dualise, and this even simplifies understanding of topological semimetals and Kane–Mele invariants by passing to Dirac-stringy picture. ◮ A lattice Z d gives (A) unit cell and (B) Brillouin zone. These are T-dual d -tori. T-duality “mixes but preserves topology”, e.g. exchanges index maps with geometric restriction maps. ◮ New notions of crystallographic T-duality and bulk-boundary correspondence allow new index theorems to be deduced. 2 / 27
Magnetic monopole and the Hopf bundle On R 3 \ { 0 } , there is a non-trivial U (1) bundle L Hopf : H 2 ( R 3 \ { 0 } ) ∼ = H 2 ( S 2 ) ∼ = Z . Magnetic field (curvature 2-form F ) has no global vector potential A . Chern number � S 2 F ∈ Z ↔ monopole charge. Dirac string is the Poincar´ e dual description: PD H 2 ( R 3 \ { 0 } ) ∼ = H 2 ( S 3 \ { 0 , ∞} ) ∼ = H 1 ( S 3 , { 0 , ∞} ) . PD → d − 1 form ( H d − 1 ). ← Roughly: 1-submanifold ( H 1 ) Poincar´ e � S 2 with F PD ↔ intersection pairing Dirac string # S 2 . Chern pairing 3 / 27
2-band crossings and monopoles x ∈ S 2 ∼ = CP 1 via the − 1 eigenspace of spin operator Unit 3-vector � x · σ . Over S 2 , these eigenspaces assemble into L Hopf → S 2 . � A 3-vector field h specifies a family of 2 × 2 Hamiltonians: H ( k ) = h ( k ) · σ , k ∈ T . Spectrum of H ( k ) is ±| h ( k ) | , so bands cross at zero set W of h , generically a set of Weyl points in 3D. For k ∈ T \ W , Negative eigenspace of H ( k ) is h ( k ) thought of as a CP 1 element. just unit vector � Valence line bundle E → T \ W is just the pullback ∗ ( L Hopf ) , h : T \ W → S 2 ( classifying map ) E = � � h 4 / 27
Toy model of topological insulator and semimetal 2D Chern insulator: T = T 2 , W = ∅ (gap condition). h : T 2 → S 2 ) = c 1 ( E ) ∈ H 2 ( T 2 , Z ) ∼ deg ( � = Z gives Chern number. 3D Weyl semimetal (WSM): T = T 3 , W a finite set. For each w ∈ W , take a local enclosing sphere S 2 w . Local obstruction to opening a gap at w is Ind h ( w ) := deg ( � h | S 2 w ) = c 1 ( E| S 2 w ). Poincar´ e–Hopf theorem imposes global constraint: � Ind h ( w ) = χ ( T 3 ) = 0 , ∀ vector fields h over T 3 . w ∈ W So Weyl points occur in cancelling pairs (cf. Nielsen–Ninomiya). 5 / 27
Weyl semimetal (L) S.-Y. Xu et al, Discovery of a Weyl Fermion semimetal and topological Fermi arcs, Science 349 613 (2015); (R) [—] Discovery of a Weyl fermion state with Fermi arcs in niobium arsenide, Nature Phys. 11 748 (2015). Dirac string is “invisible”, but there must be one. In solid state physics, 3D Weyl semimetals are characterised by bulk Dirac strings, which are “holographically” detected on a boundary. 6 / 27
Surface Fermi arcs detect global WSM topology [L] and [R] have topologically distinct Dirac strings in “dual picture”. In “Berry curvature picture”, their valence bundles have different distributions of Chern numbers on 2D slices. Boundary state appears for slices with nonzero Chern number ↔ Dirac string intersects the slice. Fermi arc is projected Dirac string. 7 / 27
Dirac string indicates “topological phase transition” Create a ± pair locally, stretch Dirac string around a non-trivial cycle and annihilate ± . This produces a transition from trivial insulator to weak Chern insulator, recorded by residual Dirac string (a loop). “Singular homotopy” classes of nonsingular vector fields on T are classified by H 1 ( T ), i.e. Dirac strings. These were called Euler structures by Turaev ’89. 8 / 27
� � � � � � � � Differential topology of semimetals [Mathai+T, CMP ’17] TI Weyl pt total Chern WSM charges charge classes invariants � �� � � �� � � �� � sum � � �� � restrict restrict H 2 ( T 3 \ W ) bands � � 0 → · · · H 2 ( T 3 ) H 2 ( S 2 H 3 ( T 3 ) · · · → 0 W ) bands PD PD PD PD ∂ � 0 → · · · H 1 ( T 3 ) H 1 ( T 3 , W ) � H 0 ( W ) � H 0 ( T 3 ) · · · → 0 � �� � � �� � � �� � � �� � closed Dirac strings Weyl pt total Dirac strings charges charge Dirac strings keep track of Weyl point “history”. Projection onto Fermi arcs is Poincar´ e dual to “integrating out transverse momenta” (a Gysin map), which is also a slice-wise analytic index. 9 / 27
Time-reversal Actually nature is more subtle — good experimental examples of TI and WSM have time-reversal symmetry Θ with Θ 2 = − 1. Time-reversal also implements momentum k z reversal θ : k �→ − k . If T is unit complex π numbers e ik with complex conjugation fixing k = 0 , π , then T d has 2 d fixed 0 points. k y π 0 k x Fu–Kane–Mele used “Berry curvature picture” to derive three weak Z 2 invariants ν i and one strong Z 2 invariant ν 0 in 3D. There is an easy derivation using θ -symmetric Dirac strings, which furthermore clarifies the “phase transitions”! 10 / 27
θ -symmetric Dirac strings With Θ symmetry, a pair w + , w − must have a partner pair at θ ( w + ) , θ ( w − ). A strong FKM invariant is generated by circular θ -symmetric Weyl point creation-annihilation process [Halasz–Balents ’12, PRB] . k z π ± ⊕ ⊖ ± ± 0 ⊖ ⊕ ± k x k y π 0 This suggests a homology classification of 3D TI, in terms of closed “ θ -symmetric Dirac strings” avoiding the fixed points. 11 / 27
θ symmetric Dirac strings Here are some θ -symmetric Dirac strings. k z π k z π 0 0 l 0 l y k y k y π π 0 k x 0 k x Can θ -symmetrically rotate l x , l y , l z , and l 0 onto their oppositely-oriented versions ⇒ 2-torsion cycles! These are the only independent generators since, e.g. l (0 , 0 , 0) + l z = l (0 , 0 ,π ) . = H 1 ( T 3 \ F ) 2 ∼ PD Technically, Z 4 → H 2 Z 2 ( T 3 , F ; Z (1)) ← RHS is the cohomological meaning of FKM invariants [De Nittis–Gomi ’16 CMP] . 12 / 27
θ -symmetric Dirac strings and Z 2 -monopoles Fermi arcs and Dirac cones ( ν 0 ) can transmute! k z w + θ ( w − ) ⊕ ⊕ ⊕ ⊕ ⊖ ⊖ ⊖ ⊖ w 1 ⊖ ⊖ ⊖ ⊖ ⊕ ⊕ ⊕ ⊕ θ ( w + ) k x k y Trap w + , θ ( w + ) between two k z θ -symmetric 2-tori π ⊕ ⊖ (purple/blue). Dirac string only 0 ⊖ pierces blue torus: ⊕ ν blue = − ν purple . k y π 0 k x Weyl points are “ Z 2 -FKM monopoles” [T+Sato+Gomi, Nucl.Phys.B ’17] . 13 / 27
Topological phase in Su–Schrieffer–Heeger model B B B B B . . . . . . A A A A A A n = − 1 n = 0 n = 1 n = 2 n = 3 Z -translations and sublattice operator S = 1 A ⊕ − 1 B . A chiral/super-symmetric Hamiltonian H = H † commutes with Z , but H S = − S H . So H exchanges A ↔ B . After Fourier transform to L 2 ( S 1 ) ⊕ L 2 ( S 1 ), � � 0 U ( k ) H S = − S H ⇐ ⇒ H ( k ) = , U ( k ) ∈ C U ( k ) ∗ 0 ∈ spec( H ) ⇔ U ( k ) ∈ C ∗ . “Gap condition”: 0 / 14 / 27
SSH model Wind( U : S 1 → C ∗ )) distinguishes topological phases of gapped, Z -invariant, supersymmetric 1D Hamiltonians! B B B B B . . . . . . A A A A A A n = − 1 n = 0 n = 1 n = 2 n = 3 � 0 � 1 Intracell H blue ( k ) = has winding number 0. 1 0 � 0 � e i k Intercell 1 H red ( k ) = has winding number 1. e − i k 0 1 Recall that translation becomes multiplication by e i k under Fourier. 15 / 27
SSH model Wind( U : S 1 → C ∗ )) distinguishes topological phases of gapped, Z -invariant, supersymmetric 1D Hamiltonians! B B B B B . . . . . . A A A A A A n = − 1 n = 0 n = 1 n = 2 n = 3 � 0 � 1 Intracell H blue ( k ) = has winding number 0. 1 0 � 0 � e i k Intercell 1 H red ( k ) = has winding number 1. e − i k 0 1 Recall that translation becomes multiplication by e i k under Fourier. 15 / 27
SSH model Puzzle: H blue ∼ unitary H red , so how can Wind( U ) be seen?? B B B B B . . . A A A A A A n = − 1 n = 0 n = 1 n = 2 n = 3 n ′ = − 1 n ′ = 0 n ′ = 1 n ′ = 2 Fourier trans. required origin choice for each sublattice Z . A boundary “fixes the gauge”, and also cuts a red link. B B B B B . . . A A A A A A The boundary “detects” the winding invariant of H red analytically as a “dangling zero A -mode”! 16 / 27
SSH model Puzzle: H blue ∼ unitary H red , so how can Wind( U ) be seen?? B B B B B . . . A A A A A A n = − 1 n = 0 n = 1 n = 2 n = 3 n ′ = − 1 n ′ = 0 n ′ = 1 n ′ = 2 Fourier trans. required origin choice for each sublattice Z . A boundary “fixes the gauge”, and also cuts a red link. B B B B B . . . A A A A A A The boundary “detects” the winding invariant of H red analytically as a “dangling zero A -mode”! 16 / 27
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